Thomas' Calculus: Early Transcendentals

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SERIES We say that the series n =0 a n converges to L , or n =0 a n = L , in case the limit of the sequence of partial sums N n =0 a n is L ; i.e., lim N →∞ N n =0 = L . The series n =0 a n converges in case to converges to some number L ; otherwise, it diverges . From the definition, we find that the geometric series n =0 r n = 1 1 - r if | r | < 1 and diverges if | r | ≥ 1. It follows that n = k r n = r k 1 - r if | r | < 1 by factoring out the common factor r k . Tests for series with positive terms comparison If 0 a n b n and the big series n =0 b n converges, then the small series n =0 a n converges. (But if the big series diverges, this gives no information about the small series.) limit comparison (very useful) If 0 < a n , 0 < b n , and a n b n as n → ∞ (that is, lim n →∞ a n b n = 1), then the two series both converge or both diverge. Read as “is asymptotic to”, or “behaves like”. A polynomial in n behaves like the leading term as n → ∞ .
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